Is the Brauer group $\text{Br}(K)$ of a global field $K$

• an $\ell$-divisible group for some prime $\ell$? If so, what $\ell$?

• Is $\text{Br}(K)[n]$ finite, for $n$ integer?

I know from class field theory that it fits into an exact sequence

$$0\to \text{Br}(K)\to\bigoplus_v\text{Br}(K_v)\xrightarrow{\sum_v \text{inv}_v} \mathbf{Q}/\mathbf{Z}\to 0$$

with $v$ running over all places of $K$, and $K_v$ the completion of $K$ at $v$.

but I can't conclude. What am I missing?

Is the Brauer group $\text{Br}(K)$ of a global field $K$

• an $\ell$-divisible group for some prime $\ell$? If so, what $\ell$?

• Is $\text{Br}(K)[n]$ finite, for $n$ integer?

I know from class field theory that it fits into an exact sequence

$$0\to \text{Br}(K)\to\bigoplus_v\text{Br}(K_v)\xrightarrow{\sum_v \text{inv}_v} \mathbf{Q}/\mathbf{Z}\to 0$$

with $v$ running over all places of $K$, and $K_v$ the completion of $K$ at $v$.

but I can't conclude.

Thanks very much.

Is the Brauer group $\text{Br}(K)$ of a global field $K$ a divisible group?

I know from class field theory that it fits into an exact sequence

$$0\to \text{Br}(K)\to\bigoplus_v\text{Br}(K_v)\xrightarrow{\sum_v \text{inv}_v} \mathbf{Q}/\mathbf{Z}\to 0$$

with $v$ running over all places of $K$, and $K_v$ the completion of $K$ at $v$.

but I can't conclude $\text{Br}(K)$ is divisible. What am I missing?

Followup. Is $\text{Br}(K)[n]$ finite, for $n$ integer?

Is the Brauer group $\text{Br}(K)$ of a global field $K$

• an $\ell$-divisible group for some prime $\ell$? If so, what $\ell$?

• Is $\text{Br}(K)[n]$ finite, for $n$ integer?

I know from class field theory that it fits into an exact sequence

$$0\to \text{Br}(K)\to\bigoplus_v\text{Br}(K_v)\xrightarrow{\sum_v \text{inv}_v} \mathbf{Q}/\mathbf{Z}\to 0$$

with $v$ running over all places of $K$, and $K_v$ the completion of $K$ at $v$.

but I can't conclude. What am I missing?